Difference between revisions of "Homework 4 ECE438F09" - Rhea

Revision as of 08:02, 20 September 2009

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-==Scaling of the Dirac Delta (Impulse Function)==
-<math>\displaystyle\delta(\alpha f)=\frac{1}{\alpha}\delta(f)\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;for\;\;\alpha>0</math>
-==Mini Proof==
-<math>\int_{-\infty}^{\infty}\delta(x)dx = 1</math>
-<math>\displaystyle Let\;\;\;y=\alpha x\;\;\;\;\;\;\;\;\;\;\;\;\;dx=\frac{dy}{\alpha}</math>
-<math>\displaystyle\int_{-\infty}^{\infty}\delta(\alpha x)dx=\int_{-\infty}^{\infty}\delta(y)\frac{dy}{\alpha}=\frac{1}{\alpha}</math>
-==Hence,==
-<math>\displaystyle\delta(\omega)=\delta(\frac{f}{2\pi})=2\pi\delta(f)</math>
-==Which also means that..==
-<math>P_T(f)=\frac{1}{T_s}\sum_{n=-\infty}^{\infty}\delta(f-\frac{n}{T_s})</math>
-<math>P_T(\omega)=\frac{2\pi}{T_s}\sum_{n=-\infty}^{\infty}\delta(w-n\frac{2\pi}{T_s})</math>